Preprint No. MPIMD/13-05

Author(s): Peter Benner, Patrick Kürschner

Email: kuerschner@mpi-magdeburg.mpg.de

Date: 2013-05-03

Abstract:

We investigate the factored alternating directions implicit (ADI) iteration for
large and sparse Sylvester
equations. A novel low-rank expression for the associated Sylvester residual is
established which enables cheap computations of the residual norm along the
iteration, and which yields a reformulated factored ADI iteration. The
application to generalized Sylvester equations is
considered as well.
We also discuss the efficient handling of complex shift parameters and
reveal interconnections between the ADI iterates w.r.t. those complex
shifts. This yields a further modification of the factored ADI
iteration which employs only an absolutely necessary amount of complex
arithmetic operations and storage, and which produces low-rank
solution factors consisting of entirely real data.
Certain linear matrix equations, such as, e.g., cross Gramian Sylvester, and
Stein equations, are in fact special cases of generalized
Sylvester equations and we show how specially tailored low-rank ADI iterations
can be deduced from the generalized factored ADI iteration.